Macroeconomic Volatility and Income Inequality in a Stochastically Growing Economy

Cecilia García Peñalosa GREQAM and CNRS

Stephen J. Turnovsky

University of Washington

Abstract

This paper employs a stochastic growth model to analyze the effect of macroeconomic volatility on the relationship between income distribution and growth. We initially characterize the equilibrium and show how the distribution of income depends upon two factors: the initial distribution of capital, and the equilibrium labor supply and find that an increase in volatility raises the mean growth rate and income inequality. We calibrate the model using standard parameter estimates, and find that it successfully replicates key growth and distributional characteristics. The latter part of the paper uses this framework to analyze the design of tax policy to achieve desired growth, distribution, and welfare objectives. Two general conclusions clearly emerge. First, increasing (average) welfare and the growth rate does not necessarily entail an increase in inequality. Second, fiscal policy often has conflicting effects on the distributions of gross and net income, with changes in factor prices resulting in greater pre-tax inequality but the redistributive effect of taxes yielding a more equal post-tax distribution.

November 2003

This paper has benefited from comments received from seminar presentations at Nuffield College, Oxford, and the University of Munich, the 2003 EEA meetings, and the 2003 IAES meetings.

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1. Introduction

The relationship between income inequality and economic growth is both important and

controversial. Knowledge of how these two variables are related is essential if a policymaker is to

be able to neutralize any undesired consequences of, say, a growth-enhancing policy on the

economy’s distribution of income. Its controversy derives from the fact that it has been difficult to

reconcile the different theories determining the relationship, especially since the empirical evidence

has not been able to provide clear support for one or another approach.1 Despite the controversy,

one thing is clear. Both the degree of income inequality and the growth rate are endogenous

variables and thus need to be jointly determined as equilibrium outcomes within a consistently

formulated macroeconomic growth model.

Empirical evidence suggests that macroeconomic volatility is potentially an important

channel through which income inequality and growth may be mutually related. For example, it is

well-known that the middle-income Latin American economies are associated with much greater

income inequality than are the East-Asian “tigers”. In 1990, the Gini coefficients of the distribution

of income in Brazil, Chile, Mexico, and Venezuela ranged between 55-64%, while those of Hong

Kong, Korea, Taiwan, and Singapore, were between 30 and 41%. At the same time, the former were

subject to much greater fluctuations in their respective growth rates than were the latter: during the

1980s, the standard deviation of the rate of output growth was, on average, 5.9% for the four Latin

American economies, and 2.8% for the East Asian countries. 2 In fact, using a broader set of data,

Breen and García-Peñalosa (2002) obtain a positive relationship between a country’s volatility

(measured by the standard deviation of the rate of GDP growth) and income inequality. On the other

hand, simple growth regressions have suggested that the mean growth rate is related to its volatility,

although the results are not uniform with respect to the direction of this relationship.3

Our objective in this paper is to model a mechanism through which aggregate risk -- which 1 See, among others, Stiglitz (1969), Bourguignon (1981), Aghion and Bolton (1997) and Galor and Tsiddon (1997), for theoretical work; Alesina and Rodrik (1994), Persson and Tabellini (1994), and Forbes (2000), for empirical analyses; and Aghion, Caroli, and García-Peñalosa (1999) for an overview. 2 See Breen and García-Peñalosa (2002) for a description of the data sources used for these calculations. 3 For example, using a sample of 92 countries, Ramey and Ramey (1995) obtained a negative relationship between inequality and growth, in contrast to earlier work by Kormendi and Mequire (1985) who obtained a positive relationship.

2

we shall attribute to production shocks -- has a direct impact on distribution. We employ an

extension of the stochastic growth model developed by Grinols and Turnovsky (1993, 1998), Smith

(1996), Corsetti (1997) and Turnovsky (2000b). This is a one-sector growth model in which, due to

the presence of an externality stemming from the aggregate capital stock, equilibrium output evolves

in accordance with a stochastic AK technology. Previous studies have been unable to analyze the

impact of volatility on income distribution, as they either abstract from labor, or otherwise, assume

that agents are identical in all respects. We introduce the assumption that agents are heterogeneous

with respect to their initial endowment of capital. As a result, the labor supply responses to different

degrees of risk will induce changes in factor prices and affect the distribution of income.

Adopting this framework, the equilibrium growth rate, its volatility, and the distribution of

income become jointly determined. Our analysis proceeds in several stages. To start with, we

derive the equilibrium balanced growth path in a stochastic growth model with given tax rates. We

show how this equilibrium has a simple recursive structure. First, the mean growth rate and labor

supply (employment) are determined at the point of intersection of two tradeoff locuses that

characterize the equilibrium relationship between rates of return and product market equilibrium,

respectively. The distribution of income is shown to depend upon two factors: the initial distribution

of capital, and the equilibrium labor supply (and hence risk). We find that an increase in production

risk raises the mean growth rate, its volatility, and the degree of income inequality. We next calibrate

the model using standard parameter estimates, typical of a developed economy.

The latter part of the paper uses this framework to analyze the effect of first-best taxation. As

it is well-known, the externality associated with the capital stock implies that the competitive growth

rate is too low. The first-best can then be attained through suitable taxes and subsidies. When agents

are heterogeneous, the use of growth enhancing policies raises the question of what is the impact of

first-best policies on the distribution of income. Two general conclusions emerge from our analysis.

First, increasing (average) welfare and the growth rate does not necessarily entail an increase in

inequality. In particular, we find that fiscal policy has conflicting effects on the distributions of gross

and net income. First-best policies result in changes in factor prices that increase pre-tax inequality,

but the direct redistributive effect of taxes tends to yield a more equal post-tax distribution. Second,

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the positive correlation between volatility and distribution may disappear when first-best policies are

implemented. Greater risk, by raising the labor supply and hence the wage bill, requires a lower

wage tax, thus making the tax system more progressive. This effect can be strong enough to offset

the positive effect of risk on inequality.

Research on the macroeconomic determinants of income inequality has mainly focused on

three aspects: growth, trade, and inflation. Studies of the impact of growth on distribution range from

the studies of the impact of structural change, such as the Kuznets hypothesis, to theories of skill-

biased technical change. Based on Heckscher-Ohlin theory, international trade has been argued to be

a major determinant of distribution, and this aspect has recently acquired prominence in the debate

on the increase in inequality in a number of industrialized countries. One of the most consistently

supported empirical correlations is that between inflation and inequality, explained by the fact that

because inflation is a regressive tax, it generates greater inequality. 4 Our paper seeks to introduce a

new and so far ignored factor, the degree of risk in an economy, into the analysis of inequality.

The paper is related to Alesina and Rodrik (1994), Persson and Tabellini (1994), and Bertola

(1993), who develop (non-stochastic) AK growth models in which agents differ in their initial stocks

of capital. The first two papers have, however, a very different focus as they take initial inequality

as given and examine how it affects the rate of growth. Bertola (1993) is closer to our approach in

that he emphasizes how technological parameters, specifically the productivity of capital, jointly

determine distribution and growth. He also examines how policies directed at increasing the growth

rate affect the distribution of consumption, although his assumption of a constant labor supply

implies that the distribution of income is independent of policy choices. Our approach shares with

these three papers an important limitation, namely, that the assumption that agents differ only in

their initial stocks of capital implies that there are no income dynamics. A more general study of

heterogeneity and the dynamics of distribution in growth models can be found in Caselli and

Ventura (2000).

The paper closest to our work, at least in spirit, is Aghion, Banerjee, and Piketty (1999), who

find that greater inequality is associated with more volatility. They show how combining capital

4 See Okun (1971) and Taylor (1981), and more recently, Albanesi (2002).

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market imperfections with inequality in a two-sector model can generate endogenous fluctuations in

output and investment. In their model it is unequal access to investment opportunities and the gap

between the returns to investment in the modern and the traditional sectors that cause fluctuations.

We reverse the focus, examining how exogenous production uncertainty determines output volatility

and income distribution.

The paper is organized as follows. Section 2 presents the model and derives the equilibrium

growth rate, labor supply, and volatility. Section 3 examines the determinants of the distribution of

income. Section 4 shows, analytically and numerically, that in the absence of taxation greater risk is

associated with a more unequal distribution of income. Section 5 starts by obtaining the first-best

optimum, and shows that the competitive growth rate is too low. It is followed by an analysis of

first-best taxation. Numerical analysis is then used to illustrate the distributional implications of the

various policies. Section 6 concludes, while technical details are provided in the Appendix.

2. The Model

2.1 Description of the economy

Technology and factor payments

Firms shall be indexed by j. We assume that the representative firm produces output in

accordance with the stochastic Cobb-Douglas production function

1( ) ( )j j jdY A L K K dt duα α−= + (1a)

where Kj denotes the individual firm’s capital stock, L j denotes the individual firm’s employment of

labor, K is the average stock of capital in the economy, so that L jK measures the efficiency units of

labor employed by the firm; see e.g. Corsetti (1997). The stochastic variable is temporally

independent, with mean zero and variance σ 2dt over the instant dt. The stochastic production

function exhibits constant returns to scale in the private factors -- labor and the private capital stock.

All firms face identical production conditions and are subject to the same realization of an

economy-wide stochastic shock. Hence they will all choose the same level of employment and

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capital stock. That is, K j = K and L j = L for all j, where L is the average economy-wide level of

employment. The average capital stock yields an externality such that in equilibrium the aggregate

(average) production function is linear in the aggregate capital stock, as in Romer (1986), namely

( ) ( ) ( )dY AL K dt du L K dt duα= + ≡ Ω + (1b)

where ( )L ALαΩ ≡ and / 0.L∂Ω ∂ >

We assume that the wage rate, z, over the period (t, t + dt) is determined at the start of the

period and is set equal to the expected marginal physical product of labor over that period. The total

rate of return to labor over the same interval is thus specified nonstochastically by

,j jj K K L L

FdZ zdt dtL

= =

∂= = ∂

. (2a)

where 1z L K wKα −= Ω ≡ .

The private rate of return to capital, dR, over the interval (t,t+dt) is thus determined residually by

KdY LdZdR rdt du

K−

= ≡ + (2b)

where ,

(1 )j K K L L

FrK

α= =

∂≡ = − Ω ∂

, and Kdu du≡ Ω .

These two equations assume that the wage rate, z, is fixed over the time period (t,t+dt), so

that the return on capital absorbs all output fluctuations. The rationale for this assumption is that in

industrial economies wages are usually fixed ex ante, while the return to capital is, at least in part,

determined ex post and thus absorbs most of the fluctuations in profitability.5 Differentiating the

production function and given that firms are identical, we find that the equilibrium return to capital

is independent of the stock of capital while the wage rate is proportional to the average stock of

capital, and therefore grows with the economy. 6 In addition, we have / 0r L∂ ∂ > and / 0w L∂ ∂ < , 5 In the United States, for example, the relative variability of stock returns over the period 1955-1995 were around 32% per annum, while the relative variability of wages over that same period was only 2%. 6 Intuitively, in a growing economy, with the labor supply fixed, the higher income earned by labor is reflected in higher returns, whereas with capital growing at the same rate as output, returns to capital remain constant.

6

reflecting the fact that more employment raises the productivity of capital but lowers that of labor.

Consumers

There is a mass 1 of infinitely-lived agents in the economy. Consumers are indexed by i and

are identical in all respects except for their initial stock of capital, Ki0. Since the economy grows, we

will be interested in the share of individual i in the total stock of capital, ki, defined as i ik K K≡ ,

where K is the aggregate (or average) stock. Relative capital has a distribution function ( )iG k ,

mean 1iik =∑ , and variance 2

kσ .

All agents are endowed with a unit of time that can be allocated either to leisure, li or to

work, 1− li ≡ Li . A typical consumer maximizes expected lifetime utility, assumed to be a function

of both consumption and the amount of leisure time, in accordance with the isoelastic utility function

( )0 0

1max ( ) , with 1, 0, 1 ti iE C t l e dt

γη β γ η γη

γ∞ − − ∞ < < > <∫ (3)

where 1− γ equals the coefficient of relative risk aversion. Empirical evidence suggests that this is

relatively large, certainly well in excess of unity, so that we shall assume γ < 0.7 The parameter η

represents the elasticity of leisure in utility. This maximization is subject to the agent’s capital

accumulation constraint

(1 ) (1 ) (1 ) (1 ) (1 ) ( ) ( ( ))

i k i k i K w i

C i i i i

dK rK dt K du w l KdtC dt sE dK s dK Ed K

τ τ ττ

′= − + − + − −′− + + + −

with duK = Ωdu . The fiscal instruments used by the government are a subsidy on investment in

physical capital, at rates s and ′ s for the deterministic and the stochastic component of investment,

respectively; a consumption tax, τc ; a wage tax, τw ; a tax on the deterministic component of capital

income, τk , and a tax on the stochastic component of capital income, ′ τ k . Taking expectations of this

expression and substituting back for E(dKi) , we can express this budget constraint as

(1 ) (1 ) (1 ) (1 ) 11 1

k i w i c i ki i K

rK w l K CdK dt K dus s

τ τ τ τ ′− + − − − + −= +

′− − (4a)

7 Some of the empirical estimates supporting this assumption are noted in Section 4.2 below.

7

It is important to observe that with the equilibrium wage rate being tied to the aggregate

stock of capital, the rate of accumulation of the individual’s capital stock depends on the aggregate

stock of capital, which in turn evolves according to

( )(1 ) (1 ) (1 ) (1 ) 11 1

k w c kK

r w l K CdK dt K du

s sτ τ τ τ− + − − − + ′−

= +′− −

(4b)

where l denotes the average (aggregate) fraction of time devoted to leisure. The agent therefore

needs to take this relationship into account in performing her optimization.

Government policy

The government balances the public budget each period, implying

[ ]1( ) (1 )1

kK c k w k KsE dK dt s Kdu C rK w l K dt Kdu

sτ τ τ τ τ

′−′ ′+ = + + − +′−

(5)

Note that both expenditures and tax receipts have a deterministic and a stochastic component.

Equating them respectively yields the following constraints required to maintain a balanced budget:

k sτ ′ ′= , (6a)

(1 )k w cCr w l sK

τ τ τ ψ+ − + = . (6b)

Two points should be noted. First, some of the taxes may be negative, in which case they

become subsidies, in addition to the investment subsidy. Second, the assumption that the

deterministic and stochastic components of income are taxed at different rates requires that the agent

(and the tax authority) disentangle the deterministic from the stochastic components of income,

something that may not be unlikely in practice.8 However, this assumption is mainly made for

analytical simplicity. Taxing both components at the same rate, with the government using public

debt in order to compensate any surplus or deficit, would not change any of our results, since as we

will see below, the tax rates on the stochastic component of capital income does not affect any of the

8 One rationale for differential taxes on the deterministic and stochastic components of capital income is to identify the latter with unanticipated capital gains, which typically, are taxed at a different rate.

8

equilibrium relationships.9

2.2 Consumer optimization

The consumer’s formal optimization problem is to maximize (3) subject to equations (4a)

and (4b). The first-order conditions with respect to consumption and leisure yield

( ) 111 i

ci i K

i

C l XC s

γη τ+=

− (7a)

( ) 11 i

wi i KC l wKX

l sγη τη −

=−

(7b)

where X(Ki,K) is the value function and XKi its derivative with respect to Ki (see Appendix).

In the Appendix we show that utility maximization implies that the dynamic evolution of the

stock of capital of agent i is given by

2 2(1 ) /(1 )1 2

i k

i

dK r s dt du dt duK

τ β γ σ ψγ

− − − = − Ω + Ω ≡ + Ω −

, (8)

where Ω and r are defined in equations (1) and (2). There we have expressed them as functions of

equilibrium employment, L , but assuming that the aggregate labor market clears, yields

1 (1 )j ij iL L l l= = − = −∑ ∑ (9)

and we can equally well write Ω and r as functions of (1− l).10 From (8) we see that the rate of

growth of capital -- and therefore of output -- has a deterministic and a stochastic component, so that

the average growth rate, ψ , and its standard deviation, σψ are respectively

2 2

111 2

krs

τ βγψ σ

γ

− − − = − Ω−

and ψσ σ= Ω (10)

Observe that the only difference between agents, namely their initial stock of capital, does

not appear in this equation. Hence all individuals choose the same rate of growth of their stock of 9 See Turnovsky (2000b) and García Peñalosa and Turnovsky (2002). 10 Thus we may write Ω(l) = A(1− l)α and r = (1− α)Ω(l ) , where ′ Ω (l) < 0 .

9

capital, ψ . This has two implications. First, the aggregate rate of growth of capital is identical to

the individual rate of growth and unaffected by the initial distribution of endowments, hence

dK K dt duψ= + Ω . (8’)

Second, since the capital stock of all agents grows at the same rate, the distribution of capital

endowments does not change over time. That is, at any point in time, the wealth share of agent i, ki ,

is given by her initial share ki,0 , say.

Dividing equation (7a) by (7b), we obtain the consumption to capital ratio of agent i,

11

i w i

i c i

C lwK k

τη τ

−=

+, (11)

Aggregating over the individuals and noting that kii∑ =1, lii∑ = l , the aggregate economy-wide

consumption-capital ratio is

11

w

c

C w lK

τη τ

−=

+ (11’)

In addition, the following transversality condition must hold

lim ( ) 0tit

E K t eγ β−

→∞ = (12)

With Ki( t) evolving in accordance with the stochastic path (8), (12) can be shown to reduce to11

2 21 (1 )1 2

krs

τ γγ γ σ β− − − Ω < −

which, when combined with (10), can be shown to be equivalent to the condition

11

krs

τ ψ− > − (13)

i.e. the equilibrium rate of return on capital must exceed the equilibrium growth rate. Dividing the

aggregate accumulation equation, (4b), by K , this condition can also be shown to be equivalent to

11 See Turnovsky (2000b).

10

(1 ) (1 ) (1 )c wC w lK

τ τ+ > − − (13’)

implying that part of income from capital is consumed.12. Combining with (9’), this can be further

expressed as

1

l ηη

>+

. (13”)

Recalling the individual budget constraint, (4a), we can write the individual’s mean rate of

capital accumulation as

( / ) 1 1 111 1 1

i i w i C iK

i i

E dK K l Cr wdt s s k s K

τ ττ − − +−= + −

− − − (14)

Together with equation (11), this expression implies that agent i’s supply of labor is

(1 ) /(1 )1 11 11 1

ki i

w

r ssl kw

τ ψηη τ

− − −−− = − + −

(15)

Noting the transversality condition, ψτ >−− )1/()1( sr k , (15) implies that an increase in the agent’s

capital (wealth) has a negative effect on her labor supply; wealthier individuals chose to “buy” more

leisure. In effect, they compensate for their larger capital endowment, and the higher growth rate it

would support, by providing less labor and having an exactly offsetting effect on the growth rate.

Because the rate of growth is the same for all agents, individual labor supplies are linear in the wealth shares of agents. The aggregate labor supply, 1 1 ii

l l− = − ∑ , is then independent of the

initial distribution of capital. Summing equation (15) over the agents and using the fact that 1ii

k =∑ , we obtain the aggregate labor supply relation,

(1 ) /(1 )1 11 11 1

k

w

r sslw

τ ψηη τ

− − −−− = − + −

, (15’)

and combining (15) and (15’) we can derive the following expression for the “relative labor supply”

12 This latter condition reduces to C K > 0 in the original Merton (1969) model, which abstracted from labor income.

11

( )11i il l l kη

η

− = − − + (15”)

Again we see that the transversality condition, now expressed as (13”), implies a positive

relationship between relative wealth and leisure.

2.3. Macroeconomic equilibrium

The key equilibrium relationships can be summarized by

Equilibrium growth rate

2 2

111 2

krs

τ βγψ σ

γ

− − − = − Ω−

(16a)

Equilibrium Volatility

ψσ σ= Ω (16b)

Individual consumption-capital ratio

11

i w i

i c i

C lwK k

τη τ

−=

+ (16c)

Aggregate consumption-capital ratio

11

w

c

C w lK

τη τ

−=

+ (16d)

Individual Budget Constraint

1 1 1 11 1 1

k w i c i

i i

l Cr ws s k s K

τ τ τψ − − − += + −

− − − (16e)

Goods market equilibrium

CK

ψ = Ω − (16f)

12

Government budget constraint

(1 )k w cCr w l sK

τ τ τ ψ+ − + = (16g)

Recalling the definitions of )(lr , )(lw , and Ω(l), and given ki , these equations jointly

determine the individual and aggregate consumption-capital ratios, i iC K , C K , the individual and

aggregate leisure times, li , l, average growth rate, ψ , volatility of the growth rate, σψ , and one of

the fiscal instruments given the other three policy parameters. Note that the tax and subsidy on the

stochastic components of investment and the return to capital, have no effect on the equilibrium

variables and thus ′ τ k = ′ s can be set arbitrarily.

Using (16a), (16d), and (16f), the macroeconomic equilibrium of the economy can be

summarized by the following pair of equations that jointly determine the equilibrium mean growth

rate, ψ , and average leisure l:

RR 2 2(1 ) ( )(1 ) /(1 )1 2

kl sα τ β γψ σγ

− Ω − − −= − Ω

−, (17a)

PP 1( ) 11 1

w

c

lll

ταψη τ

−= Ω − + −

. (17b)

The first equation describes the relationship between ψ and l that ensures the equality

between the risk-adjusted rate of return to capital and return to consumption. The second describes

the combinations of the mean growth and leisure that ensure product market equilibrium holds.

2.4. The laissez-faire economy

It is convenient to examine the equilibrium in the absence of taxation. Setting all taxes and

subsidies to zero, the equilibrium mean growth rate and leisure are determined by the following pair

of equations:

RR: 2 2(1 ) ( ) ( )1 2

l lα β γψ σγ

− Ω −= − Ω

−,

13

PP: ( ) 11

lll

αψη

= Ω − −

,

The laissez-faire RR and PP locuses are depicted in Figure 1, and their formal properties are

derived in the Appendix.13 First, note that equation PP is always decreasing in l, reflecting the fact

that more leisure time reduces output, thus increasing the consumption-output ratio and having an

adverse effect on the growth rate of capital. On the other hand, for RR we have

21 ( ) ( )1

l ll

ψ α γ σγ

∂ − ′= − Ω Ω ∂ −

This expression is unambiguously negative for γ < 0, as the empirical evidence suggests, and the

case that we shall assume prevails. Intuitively, a higher fraction of time devoted to leisure reduces

the productivity of capital, requiring a fall in the return to consumption. This is obtained if the

growth of the marginal utility of consumption rises, that is, if the balanced growth rate falls. Under

plausible conditions, the two schedules are concave, and an equilibrium exists if

212

AAβ γα γ γ σ−

− + > − .

We will see in our numerical calibrations that this condition is met for reasonable parameter values.

3. The Determinants of the Distribution of Income

In order to examine the effect of risk on income distribution, we consider the expected

relative income of an individual with capital ki . Her (expected) gross income is simply

( ) (1 )i i iE dY rK wK l= + − , while expected average income is ( ) (1 )E dY rK wK l= + − . Using

equation (15) to substitute for labor, we can express the relative (expected) income of individual i,

( ) ( )i iy E dY E dY≡ , as

( , ) (1 ) (1 )(1 ) (1 )(1 )i i i i i i

wy l k k k k kl

αη η

= + − = + −+ Ω + −

(18)

which we may write more compactly as:

13 See also Turnovsky (2000b).

14

( , ) 1 ( )(1 ), where ( ) 1(1 )(1 )i i iy l k l k l

lαρ ρ

η= − − ≡ −

+ −, (18’)

Equation (18’) emphasizes that the distribution of income depends upon two factors, the

initial (unchanging) distribution of capital, and the equilibrium allocation of time between labor and

leisure, insofar as this determines factor rewards. The net effect of an increase in initial wealth on

the relative income of agent i is given by ρ(l). As long as the equilibrium is one of positive growth,

it is straightforward to show that14

0 ( ) 1lρ< < (19)

Thus relative income is strictly increasing in ki , indicating that although richer individuals choose a

lower supply of labor, this effect is not strong enough to offset the impact of their higher capital

income. As a consequence, the variability of income across the agents, σy , is less than their

(unchanging) variability of capital, σk .

The second point to note is that we can rank different outcomes according to inequality

without needing any information about the underlying distribution of capital. For a given distribution

of capital, changes in risk or policy affect the distribution of income solely through their impact on

relative prices, as captured by )(lρ . Correia (1999) has shown that when agents differ only in their

endowment of one good, there exists an ordering of outcomes by income inequality, as measured by

second-order stochastic dominance.15 That ordering is determined by equilibrium prices, and is

independent of the distribution of endowments.

To illustrate this, we consider an analytically convenient measure of inequality, the standard

deviation of relative income, σy , where

DD σy = ρ(l)σ k (17c)

Given the standard deviation of capital, σk , the standard deviation of income is a decreasing and

14 Writing ( )1( ) (1 ) (1 )(1 )

(1 )(1 )l l l l

lρ η α α

η= − − + − − + −

. If the equilibrium is one of positive growth, (17b) implies

that the first term in brackets is positive, thus ensuring that ρ(l) > 0. The fact that ρ(l) < 1 is immediate from its definition. 15 Her results also require that the economy be amenable to Gorman aggregation, which is the case in our setup.

15

concave function of aggregate leisure time. This is because as leisure increases (and labor supply

declines) the wage rate rises and the return to capital falls, compressing the range of income flows

between the wealthy with large endowments of capital and the less well endowed.

The lower panel of Fig. 1 illustrates the DD locus, which can be shown to be concave in l.

Thus, having determined the equilibrium allocation of labor from the upper panels in Fig. 1, (17c)

determines the corresponding unique variability of income across agents.

Because taxes also have direct redistributive effects, we need to distinguish between the

before-tax and after-tax distribution of income. We therefore define the agent’s after-tax (or net)

relative income as

(1 ) (1 )(1 )( , , , ) 1 ( , , )(1 )(1 ) (1 )(1 )

NET NETk i w ii i k w w k i

k w

r k w ly l k l kr w l

τ ττ τ ρ τ ττ τ

− + − −≡ = − −

− + − − (20a)

where, ρNET summarizes the distribution of after-tax income and is related to corresponding before-

tax measure, ρ(l), by

( ) ( )( , , ) ( ) 1 ( ) (1 )(1 ) (1 )(1 )

NET w kw k

w k

l l l τ τρ τ τ ρ ρ αα τ α τ

−= + − −

− + − − (20b)

with the standard deviation of after-tax income given by

( , , )NET NETy w k klσ ρ τ τ σ= (20c)

From (20a) and (20b) we see that fiscal policy exerts two effects on the after-tax income distribution.

First, by influencing gross factor returns it influences the equilibrium supply of labor, l, and

therefore the before-tax distribution of income, as summarized by ρ(l). In addition, it has a direct

redistributive effect, which is summarized by the second term on the right hand side of (20b). The

dispersion of pre-tax income across agents will exceed the after-tax dispersion if and only if τk > τw .

As we will see below, in most cases tax increases affect the before-tax and after-tax distributions in

opposite ways.

Lastly, we compute individual welfare. By definition, this equals the value function used to

solve the intertemporal optimization problem evaluated along the equilibrium stochastic growth

16

path. For the constant elasticity utility function, the optimized level of utility for an agent starting

from an initial stock of capital, Ki,0 , can be expressed as

( )( )0 ,02

( / )1( )1/ 2( 1)

i i ii i

C K lX K K

γηγ

γ β γ ψ γ σ=

− + − (21)

The welfare of individual i relative to that of the individual with average wealth is then

( )( )

(1 )

( ) ,i i i ii i

C K l lx k kl lC K

η γγ ηγγ

γ ηγ

+

= =

(22)

where the second term has been obtained by substituting for the consumption-capital ratio. Using

equations (15), we can express relative welfare as

(1 )1( )

1i

i ikx k k

l

γ ηη

η

+ −

= + + . (22’)

Consider now two individuals having relative endowments k2 > k1. Individual 2 will have

both a higher mean income but also higher volatility. The transversality condition (13”) implies that

if γ > 0, then their relative welfare satisfies x(k2) > x(k1) > 0 , while if γ < 0, x(k1) > x(k2) > 0 .

However, in the latter case absolute welfare, as expressed by (19) is negative. Thus in either case,

the better endowed agent will have the higher absolute level of welfare, so that the distribution of

welfare moves together with that of income.

4. The Relationship between Volatility and Inequality

We now turn to the relationship between volatility, growth, and the distribution of income,

focusing on how these relationships respond to an increase the volatility of production, σ 2 . In this

section we examine the case of an economy without taxation. We discuss the relationship

analytically and then supplement this with some numerical simulations.

4.1. Analytical properties

The effect of risk operates through its impact on the incentives to accumulate capital. An

17

increase in 2σ shifts the RR curve only, and for γ < 0, it shifts the RR curve upwards, as seen in Fig.

2. Given the fraction of time devoted to leisure, the shift in RR tends to increase the growth rate. The

higher ψ increases the return to consumption, which raises the labor supply, and hence the return to

capital relative to that of consumption, causing a further increase in the growth rate. In addition greater risk increases the variance of the growth rate, 222 σσψ Ω= , because of both the direct effect

of σ 2 and the indirect impact of a lower l on Ω .

From equation (18’) the effect of an increase in risk on the relative gross income of an agent

with capital share ki is given by

( )2 2

( )sgn sgn (1 ) sgn 1ii i

dy k dl k kd l d

ρσ σ

∂ = − − = − ∂

An increase in the average leisure time raises the income share for those with a wealth share below

the average, and reduces the income share of those with wealth above average. That is, for a given

distribution of wealth, a higher equilibrium value of l reduces the relative income of individuals

with wealth (and hence income) below the mean, and increases the relative income of agents with an

above-average endowment. Consequently, inequality rises. Greater volatility of the production

shock, as measured by a higher σ 2 , increases the average growth rate and its volatility, reduces the

amount of time devoted to leisure, and increases income inequality.

The intuition for these results is straightforward. Because agents are sufficiently risk-averse,

a greater variance of output has a strong income effect that makes them increase savings.

Consequently, the growth rate increases. Note from the PP locus that the allocation of labor is

unaffected by σ 2 for a given growth rate. A higher growth rate, however, implies higher future

wages, and hence higher consumption for any extra time spent at work. It therefore reduces leisure

and the change in leisure time, in turn, affects the distribution of income. An increase in leisure time

reduces the labor supply, lowering the return to capital and increasing that to labor. Since labor is

more equally distributed than capital, the income gap between any two individuals falls. Note that

with an inelastic supply of labor, risk would not affect relative incomes. In this case, the income of

agent i would be given by yi = (w + rki) / (w + r) . With the AK technology resulting in a constant

wage and interest rate, this expression would be unaffected by risk. In our setup risk matters because

18

it affects the growth rate, and this, in turn, impacts the labor supply and factor rewards.

4.2. Calibrations

To obtain further insights into the impact of risk on the equilibrium, and in particular the

relationship between growth and income inequality, we perform some numerical analysis. We

calibrate a benchmark economy, using the following mostly conventional parameter values:

Calibration Parameters

Production 33.0 ,33.0 == αA

Taste 25.1 ,2 ,04.0 =−== ηγβ

Risk 0.05, 0.1, 0.2, 0.3, 0.4, 0.5σ =

The choice of production elasticity of labor measured in efficiency units implies that 33% of output

accrues to labor. The estimated labor share of output in industrial economies is usually much higher,

around 60%. Such a figure presents two problems. First, one consequence of the Romer technology

being assumed here, is that such a value implies an implausibly large externality from aggregate

capital. Moreover, such a high value of α would mean that the elasticity of firm output with respect

to aggregate capital is greater than with respect to own capital. Second, empirical studies using a

“broad” concept of capital that includes investment in human capital obtain a much lower output

elasticity of labor than that implied by labor share data. Hence, in line with the estimations of

Mankiw, Romer, and Weil (1992) we use a value of 33%. The choice of the scale parameter A =

0.33, is set to yield a plausible value for the equilibrium capital-output ratio.

Turning to the taste parameters, the rate of time preference of 4% is standard, while the

choice of the elasticity on leisure, 25.1=η , implies that about 75% of time is devoted to leisure,

consistent with empirical evidence. Estimates of the coefficient of relative risk aversion are more

variable throughout the literature. Values of the order of γ = −18 (and larger) have sometimes been

assumed to deal with the equity premium puzzle (see Obstfeld, 1994). However, these tend to yield

implausibly low values of the equilibrium growth rate. By contrast, real business cycle theorists

19

routinely work with logarithmic utility functions (γ = 0) . More recently, a consensus seems to be

emerging of values between 2 and 5 (see Constantinides, Donaldson, and Mehra 2002) and our

choice of 2γ = − (coefficient of relative risk aversion = 3) is well within that range.

Our main focus is on considering increases in exogenous production risk, which we let vary

between σ = 0.05 and 5.0=σ . The value σ = 0.05 is close to the mean for OECD countries

considered by Gali (1994) and Gavin and Hausmann (1995). Gavin and Hausmann present estimates

for a wide range of countries and σ = 0.10 corresponds to countries subject to medium production

risk. The highest value in our calibrations has been chosen so as to yield a standard deviation of the

rate of growth of around 10%, in line with the estimates for high-volatility economies (see Ramey

and Ramey, 1995).

Table 1 reports our assumed as well as some actual distributions. 16 The first line in Table 1

reports the distribution of wealth for the US. In 1998, the bottom 40% of the population held 0.2% of

total wealth, while the top 20% owed 83.8% of the total. The second line gives our assumed

distribution. The next two lines are the distributions of income in the US in 1997 and in Venezuela –

a high inequality, high volatility economy- in 1998. The last line reports the income distribution

generated by the model, using the hypothetical wealth distribution for the case of low risk σ = 0.05.

To obtain it we have assumed that the bottom income group has no wealth and no labor endowment

(i.e, zero income). Otherwise, since wages are identical for all workers, we would have a very large

group with the same income at the bottom of the income distribution. Our assumption implies that

the income share of the two bottom groups is 13.99 and hence of a similar magnitude to that

observed in the data. The resulting Gini coefficient, 42.89, is close to recent estimates for the US.

Table 2 reports the impact of increases in the volatility of the output shock on the equilibrium

labor supply, the average rate of growth and its standard deviation, the Gini coefficient of income,

and on overall welfare. Welfare changes reported are calculated as the percentage equivalent

variations in the initial stock of capital of the average individual necessary to maintain the level of

utility following the increase in risk from the benchmark level σ = 0.05 reported in the first row.

Line 1 of the table suggests that treating σ = 0.05 as a benchmark case leads to a plausible

16 See Wolff (2001) for the distribution of wealth, and World Bank (2002, table 2.8) for those of income.

20

equilibrium, having a 3.28% mean growth rate and 1% relative standard deviation, with 76% of time

allocated to leisure. The implied distribution of income is also plausible, as noted. As risk increases

from σ = 0.05, Table 2 indicates the following. The mean growth rate increases, as does its

standard deviation. The net effect of the greater risk dominates the positive effect of the higher

growth rate, so that the increase in risk reduces average welfare. It should be noted that for the

plausible range of σ < 0.20, the welfare loss is relatively modest. This is a characteristic limitation

of this class of model having only aggregate risk, and has been discussed elsewhere in the

literature.17 More to the point here, we see that greater risk is associated with a substitution toward

more labor (less leisure), and an increase in income inequality -- as measured by the Gini coefficient

-- consistent with the formal analysis presented in Section 4.1.

In terms of magnitudes, the effect of risk on the Gini coefficient is quite modest, at least for

plausible degrees of risk, with an increase in risk from σ = 0.05 to 5.0=σ raising the Gini

coefficient by 1.5 points. It is interesting to note that income inequality in the US increased by 2.5

Gini points between 1980 and 1990, and that this has been considered a sizeable increase. Moreover,

small changes in risk may play a significant role if they give rise to large policy responses.

5. Taxation

A familiar feature of the Romer (1986) model is that by increasing the growth rate, an

investment subsidy may move the equilibrium closer to the social optimum. When agents are

heterogeneous, two questions arise. First, how to finance this subsidy if the government is concerned

about inequality as well as about average welfare. An investment subsidy raises the return to capital

and will tend to favor those with large capital holdings. If the subsidy were financed by a lump-sum

tax, the system would redistribute away from those with lower incomes to those with higher

incomes. Are there ways in which this reverse redistribution can be avoided? Second, we want to

know whether the use of first-best policies has any implication for the relationship between volatility

and inequality. In this section we investigate these questions in some detail. We begin by deriving

17 Most notably it is characteristic of Lucas’s (1987) of the cost of business cycles, and it is also discussed at more length for a model closer to present by Turnovsky (2000b).

21

the first-best optimal rate of growth and allocation of labor.

5.1. The first-best optimum

Given the externality stemming from the aggregate capital stock, finding the first-best

optimums amounts to solving the following problem:

( )0 0

1max ( ) , ti iE C t l e dt

γη β

γ∞ −∫ (23a)

subject to

( )i i i idK K C dt K du= Ω − + Ω (23b)

In the Appendix we show that the solution to this problem is given by the equations

R’R’ 2 2( ) ( )1 2l lβ γψ σ

γΩ −

= − Ω−

(17a’)

PP ( ) 11

lll

αψη

= Ω − −

(17b)

DD ( )y klσ ρ σ= (17c)

( )1

C l lK l

αη

Ω=

− (17d)

where the tilde denotes the first-best optimum. Note that the only difference with the solution to the

competitive equilibrium in the absence of taxes is that the social rate of return to capital now takes

into account the production externality and hence exceeds the private return.18 The R’R’ schedule

lies above RR. Given that the PP schedule is steeper than RR, the upward shift of RR results in a

higher growth rate, lower leisure, and therefore increases inequality, as can be seen from Figure 2.19

18 The transversality condition (12) for the central planner’s problem again reduces to (13) but is now automatically satisfied without further restrictions being imposed. 19 The reason why the social planner chooses less leisure is that there are in fact two externalities in the model. On the one hand, a greater individual stock of capital increases the aggregate level of technology. On the other, a higher labor supply raises the marginal product of capital and induces greater accumulation of capital, thus increasing the level of technology.

22

5.2 First-best taxation

Comparing the first-best optimum, described by R’R’, and PP with the decentralized

equilibrium, RR, PP we can see that the tax-subsidy system can be used to attain the optimal growth

rate, leisure time, and consumption/capital ratio by setting

1 1, i.e. 1

ww c

c

τ τ ττ

−= = −

+ , (24a)

1(1 ) 1, i.e. (1 )1

kks

sτα α α τ−

− = = + −−

, (24b)

1 1 111 1

k

w

ll

τ ητ α

− −= − − −

, (24c)

where the last equation is obtained from the government’s budget constraint, (16g). The first two

equations represent intuitive optimality conditions. The first states that any wage tax should be

offset with an equivalent consumption tax so as not to distort the leisure-consumption choice.

Interpreting the tax on wage income as a negative tax on leisure, (24a) states that the two utility

enhancing goods, consumption and leisure, should be taxed uniformly. This result is a

straightforward application of the Ramsey principle of optimal taxation; see Deaton (1981) and

Lucas and Stokey (1983). If the utility function is multiplicatively separable in c and l, as it is here,

then the uniform taxation of leisure and consumption is optimal. The second condition simply

ensures that the private rate of return on investment must equal the social return, and for this to be so

the subsidy to investment must exceed the externality by an amount that reflects any tax on capital

income. 20 Note from the third equation that unless consumption equals total income, (in which case

there is zero growth), the replication of the first optimum requires differential taxes on wages and

capital; τw >< τ k according to whether there is positive or negative growth.

From (24a) – (24c) we obtain the following solutions for the tax rates, given an arbitrarily

chosen subsidy rate, s:

20The optimal tax rates set out in (24) are similar to those obtained by Turnovsky (2000a) in a pure deterministic representative agent endogenous growth model.

23

ˆ1Ks ατ

α−

=−

(25a)

(1 )ˆ ˆ(1 )w c

l lsl l

ητ τη

− −= = −

− − (25b)

There are two things to note about these expressions. First, ˆ τ K is very sensitive to the (arbitrary)

choice of s. Second, it is possible for the wage tax to be positive and the consumption tax negative,

and vice versa, and as we will see below, both situations arise in our calibration results.

What is the impact of the first-best taxation system on distribution? Recall that the

dispersion of gross income is given by (17c), where ρ(l) is a decreasing function of leisure time.

Since the policy increases the time allocated to labor, it will increase gross income inequality. The

dispersion of net income in the decentralized economy that mimics the centrally planned equilibrium

is obtained by substituting the tax rates, (25a), (25b), into (20b) to yield

( )1( ; , ) 1

(1 )(1 ) 1 (1 ) /NET

w kll l l

αρ τ τη α η

= −+ − + − −

(26)

The striking aspect about (26) is that the distribution of net income is independent of the (arbitrary)

choice of fiscal instruments employed to achieve this objective. As long as the equilibrium is one

with positive growth, the optimal tax requires ˆ τ w < ˆ τ k . Then ρNET < ρ , and net income is less

dispersed than is gross income. In addition, in all of our simulations we find that the direct

redistributive effect of taxation dominates the indirect effect of changes in factor prices so that the

distribution of net income is less unequal than in the economy without taxes.

When the first-best tax system is implemented, the effect of risk on growth and leisure is

equivalent to that in the laissez-faire economy, as can be easily verified from equations (17). Greater

risk is therefore associated with a greater supply of labor and hence with more pre-tax inequality.

The effect of risk on after-tax inequality is, however, ambiguous. Differentiating (26) with respect

to l , we can see that there are two opposing effects. On the one hand, more leisure tends to reduce

pre-tax inequality. On the other, and as long as the subsidy rate is less than 1, a lower labor supply

implies that a higher wage tax is required in order to finance any given subsidy rate (see (25b)),

making the fiscal system less progressive. Either effect can dominate, implying that greater risk need

24

not result in a more unequal distribution of after-tax income.

5.3 Calibrations

Table 3a reports the numerical effects of implementing the first-best taxes using the

parameter values described in section 4.2. It offers a number of insights that both reinforce and

complement our analytical results. First, we see that the policy involves a substantial reduction in

leisure time (around 4 per cent), raising the growth rate by 2.5 percentage points, and only slightly

increasing volatility.21 ∆X is the increase in the welfare of the average individual, measured as the

percentage variation over that in an economy with the same level of risk in the benchmark equilibria

(i.e. those in Table 2). First-best taxation increases the welfare of the average individual in the

economy by over 4%.

The effects on income distribution are substantial. The large reduction in leisure time results

in a large increase in pre-tax inequality, of about 3.5 percentage points. However, the redistributive

effect is strong enough to offset this effect and yield an overall reduction in the Gini coefficient of

net income. Gross and net income inequality move in opposite ways. The extent of the reduction in

post-tax inequality relative to the economy without subsidies depends heavily on the degree of risk

in the economy. For the lowest risk level, the change is small, but for 5.0=σ it amounts to almost

two Gini points. The reason for this is that since greater risk is associated with a greater labor

supply, and hence a greater wage bill, the wage tax required to finance a give subsidy is lower, and

hence the fiscal system is more progressive.

Table 3a also illustrates the analytical results that the first-best equilibrium can be replicated

by a variety of tax/subsidy configurations, each of which leads to precisely the same post-tax

distribution of income. The sensitivity of the tax regime to changes in the subsidy rate is also borne

out. Consider for example the case of low risk, σ = 0.05. In the absence of a subsidy to investment,

the first best equilibrium can be sustained if income form capital and labor are subsidized at the rates

of 49% and 83%, respectively, while consumption is taxed at 83%! This is hardly a politically

viable tax structure. But the first-best equilibrium can also be attained if, more reasonably,

21 The implied percentage increase in labor supply is much larger, being of the order of 20%.

25

investment and consumption are subsidized at 60% and 27% respectively, with taxes on labor

income and capital income of 27% and 40%, respectively.

Table 3b replicates the results in table 3a with a lower coefficient of relative risk aversion

( 1.15γ = − , rather than 2− ). The table shows how, for certain parameter values, the positive

relationship between volatility and inequality may disappear when the first-best tax system is

implemented. As is always the case, greater volatility is associated with greater pre-tax inequality,

but in this particular example the distribution of after-tax income is (virtually) unchanged by the

degree of risk. Our numerical results highlight the fact that the divergence between pre- and post-

tax inequality is greater the more risky the economy is, implying that high risk economies face a

trade-off between pre- and post-tax inequality.

6. Conclusions

Stochastic shocks are a major source of income disparities, and an extensive literature has

explored how “luck” and the market’s tendency towards convergence combine to create persistent

inequality Yet this literature is concerned with idiosyncratic shocks that have no relation with

aggregate shocks. The idea that aggregate uncertainty may also affect the distribution of income

remains to be explored, and this paper is a first step in that direction.

We have used an AK stochastic growth model to show that, when agents differ in their initial

stocks of capital, greater growth volatility is associated with a more unequal distribution of income.

Greater risk tends to increase the supply of labor, reducing wages and raising the interest rate. If

capital endowments are unequally distributed, while labor endowments are not, the change in factor

prices raises the return to the factor that is the source of inequality, and the distribution of income

becomes more spread.

The endogeneity of the labor supply also implies that policies aimed at increasing the growth

rate will have distributional implications, and we have examined how these differ depending on the

particular form of the policies. In particular, we have compared financing an investment subsidy

through a capital income tax, a wage tax, or a consumption tax.

Our analysis yields two main conclusions. First, it is possible to simultaneously increase the

26

growth rate and reduce net income inequality. In many instances, we find that polices that generate

faster growth are associated with a reduction in the Gini coefficient, allowing the policymaker to

attain both efficiency and equity goals.

Second, it is often the case that fiscal policy has opposite effects on the distribution of gross

and net income. These results highlight the fact that rather than the usual tradeoff between equity

and efficiency, policymakers concerned with the distribution of income may face a tradeoff between

pre- and post-tax inequality. Moreover, the divergence between pre and post tax inequality is greater

the more risky the economy is. Understanding which type of inequalities agents and the social

planner care about becomes essential, particularly in high risk economies, and it raises the question

of whether a slightly more unequal distribution of both gross and net incomes may, in certain cases,

be a more viable policy than a huge, but offset, increase in pre-tax inequality.

P

P

R

R

Fig 1: Equilibrium Growth, Employment, and Income Distribution

Q

l

ψ

1

σy

−β1 −γ

D

D

M

P

P

R

R

Fig 2: Increase in Production Risk

Q

l

ψ

1

σy

−β1 −γ

D

D

M

R'

M'

Q'

Table 1 The Distribution of Income and Wealth

Q1 Q2 Q3 Q4 Q5 Gini US: wealth shares 0.2 4.5 11.9 83.4 Assumed wealth shares 0 0 4.0 12.0 84.0 US: income shares 5.2 10.5 15.6 22.4 46.4 40.8 Venezuela: income shares 3.0 8.2 13.8 21.8 53.2 49.5 Simulated income shares (σ = 0.05)

0 13.99 15.76 19.28 50.97 42.89

Table 2 Growth, Volatility, and the Distribution of Income

l ψ σψ ∆X Gini(y)

σ = 0.05 76.11 3.28 1.03 -- 42.89 σ = 0.1 76.07 3.31 2.06 -0.33 42.94 σ = 0.2 75.94 3.44 4.12 -1.63 43.12 σ = 0.3 75.70 3.67 6.21 -3.82 43.44 σ = 0.4 75.36 4.00 8.31 -6.89 43.89

5.0=σ 74.91 4.43 10.45 -10.86 44.46

Table 3a First-best Taxation: baseline parameters

s τw (= −τc )

τk l ψ σψ Gini(y) Gini(ny) ∆X

σ = 0.05

0 33.0 45.29 60.0

-82.77 -22.46

0 26.89

-49.25 0

18.34 40.30

73.41

5.78

1.07

46.21

42.08

4.29

σ = 0.1

0 33.0 45.38 60.0

-83.11 -22.69

0 26.76

-49.25 0

18.49 40.30

73.36

5.82

2.13

46.26

42.10

4.29

σ = 0.2

0 33.0 45.80 60.0

-84.51 -23.62

0 26.20

-49.25 0

19.11 40.30

73.18

5.97

4.27

46.45

42.18

4.28

σ = 0.3

0 33.0 46.51 60.0

-86.95 -25.26

0 25.22

-49.25 0

20.17 40.30

72.88

6.23

6.44

46.77

42.27

4.26

σ = 0.4

0 33.0 47.55 60.0

-90.66 -27.75

0 23.73

-49.25 0

21.72 40.30

72.44

6.60

8.63

47.22

42.40

4.22

5.0=σ

0 33.0 48.95 60.0

-96.00 -31.31

0 21.60

-49.25 0

23.81 40.30

71.85

7.09

10.86

47.81

42.55

4.17

Table 3b First-best Taxation: 15.1−=γ

s τw (= −τc )

τk l ψ σψ Gini(y) Gini(ny) ∆X

05.0=σ 0 0 0 74.63 4.68 1.05 44.80 44.80 - 05.0=σ 60.0 14.25 40.30 70.08 8.45 1.11 49.39 42.80 8.15 1.0=σ 60.0 14.09 40.30 70.05 8.48 2.22 49.41 42.80 8.15 2.0=σ 60.0 13.45 40.30 69.92 8.57 4.44 49.53 42.80 8.12 3.0=σ 60.0 12.34 40.30 69.67 8.75 6.68 49.71 42.81 8.08 4.0=σ 60.0 10.68 40.30 69.36 8.99 8.93 49.98 42.80 8.02 5.0=σ 60.0 8.34 40.30 68.92 9.30 11.22 40.32 42.78 7.94

A.1

Appendix

This appendix provides some of the technical details underlying the derivations of the

equilibrium conditions (8) and (16a) to (16g).

A.1 Consumer optimization

Agent i‘s stochastic maximization problem is to choose her individual consumption-capital

ratio and the fraction of time devoted to leisure to maximize expected lifetime utility

( )0 0

1max ( ) , ti iE C t l e dt

γη β

γ∞ −∫ −∞ < γ < 1,η > 0,γη <1 (A.1a)

subject to her individual capital accumulation constraint

(1 ) (1 ) (1 ) (1 )1

k i w i ci i

rK w l K CdK dt K dks

τ τ τ− + − − − += +

− (A.1b)

and the aggregate capital accumulation constraint

( )(1 ) (1 ) (1 ) (1 )1

K w cr w l K CdK dt Kdk

sτ τ τ− + − − − +

= +−

(A.1b’)

together with the economy-wide shock

11 '

kdk dusτ ′−

= Ω−

. (A.1c)

Since the agent perceives two state variables, Ki , K , we consider a value function of the form

( ), , ( , )ti iV K K t e X K Kβ−=

the differential generator of which is

[ ] 1 1 1( , , ) (1 )1 1 1 i

k c i wi i K

i

CVV K K t r K w l K Vt s s K s

τ τ τ − + −∂Ψ ≡ + − + − ∂ − − −

2 2 2 21 1 1 1 1(1 )1 1 1 2 2i i i i

k c wK K i K K K K i K K K KK

Cr w l KV K V K KV K Vs s K s

τ τ τ σ σ σ− + − + − + − + + + − − − (A.2)

A.2

The individual’s problem is to choose consumption, leisure, and the rate of capital

accumulation to maximize the Lagrangian

( )1 ( , )t ti i ie C l e X K K

γβ η β

γ− − + Ψ . (A.3)

In doing this, she takes the evolution of the aggregate variables and the externality as given. Taking

the partial derivatives with respect to Ci and li , and cancelling e−βt yields

( ) 111 i

ci i K

i

C l XC s

γη τ+=

− (A.4a)

( ) 11 i

wi i KC l wKX

l sγη τη −

=−

(A.4b)

In addition, the value function must satisfy the Bellman equation

( )1max ( , ) 0t ti i ie C l e X K K

γβ η β

γ− − + Ψ =

(A.5)

The Bellman equation is a function of two state variables, individual and aggregate capital, and

hence it is a partial differential equation in these two variables. Using equations (A.1b) and (A.1b’),

and given (A.2), the Bellman equation can be written as

( )2 2

( )1 ( )( , )

( ) ( )1 1 ( ) 02 2

i

i i i

ii i i K K

i iK K K K KK

E dK E dKC l X K K X Xdt dt

E dK E dK dK E dKX X Xdt dt dt

γη βγ

− + +

+ + + =

(A.6)

Next we take the partial derivative of the Bellman equation with respect to Ki , noting that li

is independent of Ki , while Ci is a function of Ki through the first-order condition (A.4a),

( ) 2, ,

( ) 11 ( )1i i i i i i i i i

i ki i i K K K K i K K K K i K K K

i

E dK E dKC l C X X r C X X K XC dt s dt

γη τβ σ− − + + − + + −

2 2( ) ( )1 1 ( ) 0

2 2i i i i i i i i

i iK K K K KK K K K K K KK

E dK E dK dK E dKX X KX Xdt dt dt

σ+ + + + = (A.7)

Consider now XKi= XK i

(Ki,K). Taking the stochastic differential of this quantity yields:

A.3

2 21 1( ) ( )( ) ( )2 2i i i i i i i i i iK K K i K K K K K i K KK i K KKdX X dK X dK X dK X dK dK X dK= + + + + (A.8)

Taking expected values of this expression, dividing by dt, and substituting the resulting equation

along with (A.4a) into (A.7) leads to:

2 ( )1 [ ] 01

i

i i i i

KkK i K K K K K

E dXr X K X KX

s dtτ β σ− − + + + = −

, (A.9)

The solution to this equation is by trial and error. Given the form of the objective function,

we propose a value function of the form:

2 2( , )i iX K K cK Kγ γ γ−= (A.10)

where the parameters c,γ 2 are to be determined. From (A.10) we obtain:

2 2

22 2

22 2 2 2

( ) / ; / ;

( )( 1) / ;

( ) / ; ( 1) / .

i

i i

i

K i K

K K i

K K i KK

X X K X X K

X X K

X X K K X X K

γ γ γ

γ γ γ γ

γ γ γ γ γ

= − =

= − − −

= − = −

(A.11)

We can now use equation (A.12) to re-express (A.11) as

2( ) 1 ( 1)1

i

i

K kK

K

E dXr

X dt sτβ γ σ−

= − + −−

(A.12)

Now, returning to the first-order condition (A.4a), computing the stochastic differential of

this relationship and taking expected values yields

( ) 2( ) 1( 1) ( 1)( 2)

2i

i

K i i

K i i

E dX E dC dCEX C C

γ γ γ

= − + − −

(A.13)

Along the balanced growth path, Ci Ki is constant. Hence dCi / Ci = dKi / Ki =ψdt + dw , and thus

( ) 21( 1) ( 1)( 2)

2i

i

KK

K

E dXX dt

γ ψ γ γ σ= − + − − (A.14)

As will be shown below (see equation (A.26)), the government’s balanced budget implies that the

stochastic component of the individual budget constraint is

A.4

dk du= Ω . (A.15)

Combining (A.13), (A.14), and (A.15) yields the mean growth rate of individual consumption

( ) 2 2(1 ) (1 )1 2

kr sτ β γψ σγ

− − −= − Ω

−. (A.16)

The labor supply is obtained from the first-order conditions (A.4a) and (A.4b), namely

11

wi i

c

wC Klτη τ

−=

+. (A.17)

Dividing (A.17) by K we obtain the individual consumption to wealth ratio,

11

i w i

i c i

C lwK k

τη τ

−=

+, (A.18)

and summing over all agents we have the aggregate consumption to wealth ratio,

11

w

c

C w lK

τη τ

−=

+. (A.19)

From the individual budget constraint, the rate of growth is

1 1 1 11 1 1

k w i c i

i i

l Cr ws s k s K

τ τ τψ − − − += + −

− − −, (A.20)

which using (A.18) and rearranging gives

(1 ) /(1 )1 11 .1 1 1

ki i

w

r s sl kw

τ ψηη η τ

− − − −− = −

+ + − (A.21)

A.2 The government’s budget constraint

The balanced budget implies

[ ]( ) (1 )c k K w k KsE dK dt s Kdk C r K w l K dt Kduτ τ τ τ′ ′+ = + + − + . (A.22)

Equating the deterministic and the stochastic components of (A.22) yields :

A.5

k Kdu s dkτ ′ ′= , (A.23)

(1 )k K w cCr w l sK

τ τ τ ψ+ − + = . (A.24)

Note that (A.1c) implies

′ τ k = ′ s . (A.25)

As a result, the stochastic component of individual capital accumulation is

dk du= Ω . (A.26)

A.3 Macroeconomic equilibrium

Note that the growth rate is the same for all agents, irrespective of their initial wealth

holdings. Equation (A.16) is hence the mean growth rate of the economy. The dynamic evolution of

the aggregate stock of capital is given by

( ) 2 2(1 ) (1 )1 2

kr sdK dt duK

τ β γ σγ

− − − = − Ω + Ω −

and the standard deviation (volatility) of the growth rate is

ψσ σ= Ω . (A.27)

Summing (A.21) over all agents gives a relationship between the aggregate labor supply and the

growth rate,

1 11 .1 1 1 k

slw

η φ ψη η τ

− −− = −

+ + − (A.28)

Goods market equilibrium requires ( )dK K dt du Cdt= Ω + − , which taking expectations and dividing

by K yields,

CK

ψ = Ω − . (A.29)

Equations (A.16), (A.27) (A,18), (A.19), (A.20), (A.29), and (A.24) are the macroeconomic

A.6

equilibrium conditions as specified in equations (16a) – (16g), respectively

In the absence of taxation, the equilibrium reduces to

i i

i i

C lwK kη

= , (A.30a)

2 2

1 2r β γψ σ

γ−

= − Ω−

, (A.30b)

(1 ) i

i

Cr w lK

ψ = + − − , (A.30c)

C wlK η

= , (A.30d)

ψσ σ= Ω , (A.30e)

which jointly determine the consumption-capital ratio, the average growth rate, the labor supply, and

the volatility of growth.

A.4 Existence of a balanced growth equilibrium

It suffices to focus on the economy without taxation; the introduction of taxes leads to minor

modifcations and can be analyzed analogously. Differentiating the relations in (A.30) we obtain

2( ) (1 ) ( ) 01 1RR

l ll l

ψ α α γ σγ

∂ Ω −= − − Ω < ∂ − −

, (A.31a)

( ) 1 (1 ) 0(1 ) 1PP

l ll l l

ψ α η αη

∂ Ω = − + + − < ∂ − − , (A.31b)

so that both schedules have a negative slope. Using the fact that Ω = A 1− l( )α , and under the

assumption that α <1 / 2, both the (PP) and (RR) schedules can be shown to be strictly concave (see

Turnovsky, 2000b, for more details). Also

2 2(1 )( 0) , ( 1)1 2 1

( 0) , ( 1) .

RR RR

PP PP

Al A l

l A l

α β γ βψ σ ψγ γ

ψ ψ

− −= = − = = −

− −= = = = −∞

A.7

A necessary and sufficient condition for the existence of a unique equilibrium is

ψ PP ( l = 0) >ψ RR ( l = 0). In this case, the (PP) schedule is below (RR) for l =1, and the two schedules

only cross once. This condition is satisfied when

212

AAβ γα γ γ σ−

− + > − , (A.32)

i.e. when risk is not excessively high. When equation (A.32) is not satisfied either an equilibrium

does not exists or there are two.

Note also that the PP schedule is steeper than RR if and only if

21 1 11 1

ll

η α α γ ση η γ+ − −

+ > − Ω− −

. (A.33)

Since at l = 0, l / (1− l) has its lowest and Ω(l) its highest possible value, PP is everywhere steeper

than RR if and only if

21 11

Aη α γ ση γ+ −

− > −−

. (A.34)

A.5 The centrally planned economy

The social planner’s problem (23), leads to the following Bellman equation

( )2( ) ( )1 1( , ) 0

2i i i

i i ii i i K K K

E K C E dKC l X K K X Xdt dt

γη βγ

Ω −− + + = . (A.6’)

Taking the partial derivative of this equation with respect to Ki then yields

( ) 2, ,

2

( )1 ( )

( )1 02

i i i i i i i i

i i i

ii i i K K K K i K K i K K K

i

iK K K

E dKC l C X X C X K XC dt

E dK Xdt

γη β σ− + + Ω − +

+ =

(A.7’)

and hence

2( )( ) ( 1)i

i

KK

K

E dXX dt

β γ σ= − Ω − + − , (A.13’)

A.8

which together with (A.14) and (A.15) above yield (16a’).

The first order conditions with respect to consumption and leisure, (16), together imply

i i

i

C wlK η

= , (A.19’)

Goods market equilibrium is again given by equation (A.29). Using (A.19’), the equilibrium

conditions can be expressed as

2 2( ) ( )1 2l lβ γψ σ

γΩ −

= − Ω−

, (A.35)

( ) ( ) 11

C ll lK l

αψη

= Ω − = Ω − −

. (A.36)

(A.36) is strictly decreasing and concave in l. Differentiating (A.35), we obtain

∂ψ∂l R 'R '

= −Ω( l)1− l

11−γ

− γΩ(l)σ 2

< 0 ,

22

2 2''

( ) 1 ( )(1 2 )(1 ) 1RR

l ll lψ α α γα α σ

γ ∂ Ω −

= − − Ω − ∂ − − .

(A.35) is thus decreasing in l and a sufficient condition for concavity is α < 1 / 2. The necessary and

sufficient condition for the existence of a unique equilibrium, ψ PP (l = 0) >ψ RR ' '(l = 0), is now

212

AAβ γγ γ σ−

− + > − . (A.32’’)

Note also that (A.36) schedule is steeper than (A.35) if and only if

21 11

Aη γ ση γ+

− > −−

. (A.34’)

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